Geometry and dynamics in the moduli spaces proved to be extremely efficient in the study of surface foliations, billiards in polygons and in mathematical models of statistical and solid state physics like Ehrenfest billiards or Novikov's problem on electron transport. Ideas of study of surface dynamics through geometry of moduli spaces originate in works of Thurston, Masur and Veech. Contributions of Avila, Eskin, McMullen, Mirzakhani, Kontsevich, Okounkov, Yoccoz, to mention only Fields Medal and Breakthrough Prize winners, made geometry and dynamics in the moduli spaces extremely active area of modern mathematics.  Moduli spaces of Riemann surfaces and related moduli spaces of Abelian differentials are parameterized by a genus g of the surface. Considering all associated hyperbolic (respectively flat) metrics at once, one observes more and more sophisticated diversity of geometric properties when genus grows. However, most of metrics, on the contrary, progressively share certain similarity. Here the notion of “most of” has explicit quantitative meaning, for example, in terms of the Weil-Petersson measure.  Global characteristics of the moduli spaces, like Weil-Petersson and Masur-Veech volumes, Siegel-Veech constants, intersection numbers of psi-classes were traditionally studied through algebra-geometric tools, where all formulae are exact, but very difficult to manipulate in large genus. Most of these quantities admit simple uniform large genus approximate asymptotic formulae.  I will give a survey of recent fundamental discoveries of these large genus universality phenomena and of relations between them.